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By Groves M.D., Haragus M.

This text provides a rigorous lifestyles concept for small-amplitude threedimensional traveling water waves. The hydrodynamic challenge is formulated as an infinite-dimensional Hamiltonian procedure during which an arbitrary horizontal spatial course is the timelike variable. Wave motions which are periodic in a moment, assorted horizontal course are detected utilizing a centre-manifold aid strategy in which the matter is lowered to a in the community an identical Hamiltonian process with a finite variety of levels of freedom.

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We can therefore apply the resonant version of the Lyapunov centre theorem due to Weinstein [32] and further developed by Moser [26], which states that the nonresonance condition on the eigenvalues can be replaced by the requirement that the quadratic part of the Hamiltonian is positive-definite. The following result is obtained by applying the Weinstein-Moser theorem to the reduced Hamiltonian system and to its further reduction by the symmetry S2 ; in the latter case we recover the result given by Groves [10, Theorem 5] with the nonresonance condition removed.

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Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun. Pure Appl. Math. 29, 727–747. [27] PEGO, R. , AND QUINTERO, J. 2002. A host of traveling waves in a model of threedimensional water-wave dynamics. J. Nonlinear Sci. 12, 59–83. [28] PLOTNIKOV, P. I. 1980. Solvability of the problem of spatial gravitational waves on the surface of an ideal fluid. Sov. Phys. Dokl. 25, 170–171. , AND SHINBROT, M. 1981. Three-dimensional nonlinear wave interaction in water of constant depth.

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